Statistics

Given the initial measured values Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_0} , final observed or measured values Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} and final calculated values Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} , there are several goodness of fit statistics or skill scores which can be calculated. The definition for some of the more common ones are provided below.

Brier Skill Score

The Bier Skill Score (BSS) is given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BSS = 1 - \frac{\bigg\langle \big(x_m-x_c\big)^2 \bigg\rangle}{\bigg \langle \big(x_m-x_0\big)^2 \bigg\rangle } }

(1)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_0} is the initial measured values. The BSS ranges between negative infinity and one. A BSS value of 1 indicates a perfect agreement between measured and calculated values. Scores equal to or less than 0 indicates that the mean observed value is as or more accurate than the calculated values.

Nash-Sutcliffe Coefficient

The Nash-Sutcliffe Coefficient (E) is commonly used to assess the predictive power of a model. It is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E = 1 - \frac{\bigg\langle \big(x_m-x_c\big)^2 \bigg\rangle}{\bigg\langle \big(x_m- \langle x_m \rangle \big)^2 \bigg\rangle } }

(2)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values. The Nash-Sutcliffe efficiency coefficient ranges from negative infinity to one. An efficiency of 1 corresponds to a perfect match between measured and calculated values. An efficiencies equal 0 or less indicates that the mean observed value is as or more accurate than the calculated values.

Root-Mean-Squared Error

The Root-Mean-Squared Error (RMSE) also referred to as Root-Mean-Squared Deviation (RMSD) is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle RMSE = \sqrt{ \bigg\langle \big( x_m - x_c \big)^2 \bigg\rangle } }

(3)

where where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values. The RMSE has the same units as the measured and calculated data.

Normalized-Root-Mean-Squared Error

In order to make comparing different RMSE with different units or scales (lab vs field) several non-dimensional forms of the RMSE have been proposed in literature. Here the Normalized-Root-Mean-Squared Error (NRMSE) is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle NRMSE = \frac{\sqrt{ \bigg\langle \big( x_m - x_c \big)^2 \bigg\rangle }}{\max{(x_m)}-\min{(x_m)}} }

(4)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values. The NRMSE is often expressed in units of percent. Smaller values indicate a better agreement between measured and calculated values.

Mean-Absolute Error

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle MAE = \bigg\langle \big| x_m - x_c \big| \bigg\rangle }

(5)

where where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values.

Normalized-Mean-Absolute Error

The normalized-Mean-Absolute Error is defined as

$NMAE={\frac {{\bigg \langle }{\big |}x_{m}-x_{c}{\big |}{\bigg \rangle }}{\max {(x_{m})}-\min {(x_{m})}}}$

(6)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values.

Correlation coefficient is defined as

Correlation is a measure of the strength and direction of a linear relationship between two variables. The correlation coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R } is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R = \frac { \langle x_m x_c \rangle - \langle x_m \rangle \langle x_c \rangle }{ \sqrt{ \langle x_m^2 \rangle - \langle x_m \rangle ^2} \sqrt{ \langle x_c^2 \rangle - \langle x_c \rangle ^2} } }

(7)

where where $x_{m}$ is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c } is the calculated values. A correlation of 1 indicates a perfect one-to-one linear relationship and -1 indicates a negative relationship. The square of the correlation coefficient describes how much of the variance between two variables is described by a linear fit.

Bias

The bias is a measure of the over or under estimation and is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B = \langle x_c - x_m \rangle }

(8)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values. The bias is a measure of the over or under prediction of a variable. Positive values indicate overprediction and negative values indicate underprediction.

Normalized Bias

The normalized bias is a measure of the over or under estimation and is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B = \frac{\langle x_c - x_m \rangle}{\max{(x_m)}-\min{(x_m)}} }

(9)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values. The normalized bias is a measure of the over or under prediction of a variable and is often expressed as a percentage. Positive values indicate overprediction and negative values indicate underprediction.

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